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Volume : 44

Part : 2

Proceedings of the Institute of Acoustics

 

 

Using data-driven techniques to provide feedback during material characterisation

 

Gaborit Mathieu1, Le Mans Université (LAUM, CNRS), Le Mans, France

Luc Jaouen2, Matelys, Vaulx-en-Velin, Franc

 

ABSTRACT

 

The aim of the work is to study the feasibility of using machine learning techniques to design a decision helper to assist the characterisation of acoustic materials (porous media for instance). The tool is intended to alert the human operator about specific physical phenomena occurring during the measurements or common mistakes in handling the characterization rig or its parameters. Examples of classical issues include leakage around the samples, unintentional compression during the sample mounting, errors in input parameters such as the static pressure or temperature, etc. The proposed helper relies on a physical analysis and a k-nearest neighbours classifier using the Fréchet distance to score the measurements. This approach allows to measure the similarity between curves, independently from sampling. The training phase is performed on a labelled dataset created from actual impedance tube measurements and possibly some computer generated results to bridge gaps. The inputs are frequency-dependent quantities including normal sound absorption curves, surface impedance, dynamic mass density and dynamic bulk modulus.

 

1. INTRODUCTION

 

Proper characterisation of materials (acoustic or otherwise) is hardly a perfectly transparent process. Despite standardized characterisation protocols, a number of unspoken factors come into play and can dramatically impact the results. Indeed, from the preparation of the samples to the post-processing, a lot of steps heavily rely on the human operator’s skills and experience to deliver repeatable results. The central goal of standardization committees is to factor the need for experience out of the protocols as much as possible, letting the end users of the norm with an reasonably operator-agnostic protocol.

 

In this contribution, the problem of operator experience (or lack thereof) is addressed in a different way: a proposition to use automatic classification is made, using a curve-to-curve distance metric, to give operator feedback on the samples he processed. The key idea here is to be able to detect the most common problems during the characterisation of porous media using the current protocols. This contribution is preliminary and results will be provided during the conference. The rest of the manuscript is organized as follows: the mathematical/processing framework is introduced in the next section, then the constitution of datasets is briefly exposed and finally some perspectives are presented.

 

2. CHARACTERISATION METHODS AND DATA COLLECTION

 

2.1. Dynamic properties

 

The measured data are obtained from impedance tube measurements. The 3-microphone technique as described by Iwase et al. [1, 2] (see also Fig. 1 and Fig. 2) is used to measure the dynamic mass density ρ and the dynamic bulk modulus K of porous materials in addition to their surface properties such as the normal surface impedance Z, the reflection coefficient R and the sound absorption coefficient α.

 

The protocol roughly goes as follows. A calibration procedure is performed before any measurement. Then, a sample of material is taken from a larger stock using a cutting tool (rotary cutter, band saw, abrasive thread, scissors, etc.). The key element here is to make sure that the sample fits the tube without being compressed once in place. Afterwards, the sample is placed in the tube, paying attention that no gap is left between the sample and the rigid backing and that the free surface sits perpendicular to the axis. The operator might use grease or other means to prevent air leaks around the sample if needed, still making sure not to compress the sample. The measurement is performed by exciting the tube at frequencies below the cut-off frequency of the tube and sent for post-processing to obtain the complex reflection coefficient.

 

 

Figure 1: Scheme of the experimental setup used for the measurement of the dynamic mass density and compressibility.

 

 

Figure 2: One of the Impedance tubes used to collect the experimental data. Characteristics of this tube: length ≈1.2 m, ⊘ 46 mm, f ∈ [250-4 500] Hz. Microphones 1/4”.

 

2.2. Acoustic parameters

 

From the dynamic properties ρ and K, the acoustic parameters of a material are deduced from asymptotic estimations and analytic inversions following the works by Olny & Panneton and Jaouen et al. [3–5] based on the phenomological Johnson-Champoux-Allard-Lafarge (JCAL) [6–8] model. The acoustic parameters which are determined in the direction normal to the material surface are:

  • the static air flow resistivity σ (N.s.m−4),

  • the open porosity ϕ,

  • the high frequency limit of the dynamic tortuosity α∞,

  • the viscous characteristic length Λ (m),

  • the thermal characteristic length Λ'(m),

  • the static thermal permeability k'0(m2).

 

3. MATHEMATICAL AND COMPUTATIONAL FRAMEWORK

 

In order to identify the occurrence of specific characterisation problems impacting a given result, this contribution proposes an approach based on the k-nearest neighbours (k-NN)algorithm [9, 10]. This tool allows to identify a predefined number of clusters within a dataset by measuring the distance between data points in the parameter space. Using a so-called training dataset, a classifier can be trained that can be used afterwards to assign any new data point to a given cluster. In the present case, the objects to be represented are relatively long vectors of floats which could prove difficult as the k-NN is strongly impacted by the so-called curse of dimensionality and pretty much impotent at for high-dimensional datasets. Here the training data is chosen to represent both curves from valid measurements and measurements with specific errors and mishaps.

 

In the present case an absorption results cannot really be understood as a many-dimensional vector as it physically bears only two of them : frequency and absorption amplitude. Thus measuring the distance between two absorption curves as a Euclidean distance between the vectors makes little to no sense. There are a number of ways to measure mathematical distances between curves and this contribution is based on the use of the Fréchet distance [11, 12]. One of the advantages of this option is that it can measure distances between curves having different number of points This is particularly accommodating during the classification phase as it renders the stack ever agnostic with respect to secondary measurement parameters.

 

The Fréchet distance is defined as follows: let a curve be a continuous map A : [0, 1] → S from the unit interval into S . Let two non-decreasing functions α : [0, 1] → [0, 1] and β : [0, 1] → [0, 1] representing the position on each curve (called reparameterizations) [11, 12]. The Fréchet distance between two curves A and B in S : F(A, B) is defined as:

 

 

i.e. the infimum over all reparameterizations α and β of [0, 1] of the maximum over all t ∈ [0, 1] of the distance in S between A(α(t)) and B(β(t)) with d the distance function of S .

 

4. PRELIMINARY RESULTS

 

In order to gather enough data to train the model and perform classification, experiments and post processing is still being run and more results will be presented during the conference. The production of simulated data is possible as well but the performance of the system might not be representative of what it can be on measured data as k-NN tends to lock onto local variations of the data that can occur when correlated noise pollutes the data.

 

In the meantime, a preliminary analysis was run on the data from Ref. [13] where more than 3000 of samples of the same materials were manufactured using different techniques and tested in the impedance tube by different operators. The data is composed of a 3073 absorption coefficients of 1960 points each and the preprocessing is the same as in Ref. [13] as the authors provided their code via Ref. [?]: all negative spectra are removed, a rolling filter is applied and the data can be downsampled at will. The rolling filter is used to smooth out the most prominent hiccups from the results as a machine learning approach might latch onto that in the training process.

 

The authors from Ref. [13] classified the data using 5 criteria: thickness, diameter, cutting technology, operator and sample mounting solution. The possible values for the parameters are available in Table 1.

 

Table 1: Possible classes for the different criteria. Cutting technologies acronym are explained in detail in Ref. [13]

 

 

The dataset is shuffled and divided into a training set comprising 75% of the available data and a 25% test split. Different k-NN classifiers are then trained to distinguish the different classes for each criterion in an univariate set up and the Fréchet distance is used to computed how far two spectra are from one another.

 

The results on the test split are presented in Table ?? for the whole dataset, a 50% subsampled dataset, a 75% subsampled dataset and a 90% one. The subsampling in this case can help the classifier capture the overall trend of the curve instead of its details.

 

Table 2: Success rate of the classification via Fréchet-based k-NN for the different available criteria and different sub-sampling rates.

 

 

A number of conclusions can be drawn from these preliminary results. First, note that while sub sampling usually improves machine learning classification results for vector data, it doesn’t affect significantly the results here. This could be expected: as the Fréchet distance is used, the most local variations of the spectra have little to no impact on the resulting metric. Another takeaway from these results is that the different classes in some of the criteria (the thickness for instance) are easily distinguished because they have a tremendous impact on the absorption spectra and hence are associated with a very low similarity. The operator seems to be easily distinguishable as well and so is the diameter. Finally, the cutting technology and mounting strategy do not seem to correlate well with the absorption results through the prism of the Fréchet distance.

 

These results suggest that a system based on the proposed stack is enough to give some feedback on cutting errors and thickness defects as well and uneven handling from operators. It is a first step towards wider clustering of absorption results and it shows the limits of the proposed approach on measurement data.

 

5. PERSPECTIVES

 

This preliminary study presents the first bricks of an approach to cluster absorption curves and by using a good set of training data the clusters can represent a given characterisation issue. The data is still being gathered and process and results will be presented during the conference.

 

REFERENCES

 

  1. T. Iwase, Y. Izumi, and R. Kawabata. A new measuring method for sound propagation constant by using sound tube without any air spaces back of a test material. In Proceedings of inter.noise 98, Christchurch, New Zealand, 1998.

  2. Olivier Doutres, Yacoubou Salissou, Noureddine Atalla, and Raymond Panneton. Evaluation of the acoustic and non-acoustic properties of sound absorbing materials using a three-microphone impedance tube. Applied acoustics, 71(6):506–509, 2010.

  3. R. Panneton and X. Olny. Acoustical determination of the parameters governing viscous dissipation in porous media. J. Acoust. Soc. Am., 119:2027–2040, 2006.

  4. X. Olny and R. Panneton. Acoustical determination of the parameters governing thermal dissipation in porous media. J. Acoust. Soc. Am., 123:814–824, 2008.

  5. L. Jaouen, E. Gourdon, and Ph. Glé. Estimation of all 6 parameters of johnson-champoux-allard lafarge model for acoustical porous materials from impedance tube measurements. J. Acoust. Soc. Am., 148(4):1998–2005, 2020.

  6. D. L. Johnson, J. Koplik, and R. Dashen. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J. Fluid Mech., 176:379–402, 1987.

  7. Y. Champoux and J.-F. Allard. Dynamic tortuosity and bulk modulus in air-saturated porous media. J. Appl. Phys., 70:1975–1979, 1991.

  8. D. Lafarge, P. Lemarinier, J-F. Allard, and V. Tarnow. Dynamic compressibility of air in porous structures at audible frequencies. J. Acoust. Soc. Am., 102(4):1995–2006, 1997.

  9. Evelyn Fix and Joseph L. Hodges. Discriminatory Analysis. Nonparametric Discrimination: Consistency Properties.

  10. N. S. Altman. An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression. 46(3):175–185.

  11. Helmut Alt and Michael Godau. Measuring the resemblance of polygonal curves. In Proceedings of the Eighth Annual Symposium on Computational Geometry - SCG ’92, pages 102–109. ACM Press.

  12. Efrat, Guibas, Sariel Har-Peled, Mitchell, and Murali. New Similarity Measures between Polylines with Applications to Morphing and Polygon Sweeping. 28(4):535–569. [13] Merten Stender, Christian Adams, Mathies Wedler, Antje Grebel, and Nobert Hoffmann. Explainable machine learning determines effects on the sound absorption coefficient measured in the impedance tube. 149(3):1932–1945.

 

1 mathieu.gaborit@univ-lemans.fr

2 luc.jaouen@matelys.com